Essential Physics II
E 英語で物理学の エッセンス II
Lecture 10: 30-11-15 News Schedule change:
Monday 7th December (１２月７日) NO CLASS!
Class 11/26 homework: 12/14
This week’s homework: 12/14 So far... We have seen: Electric fields, E¯ , are created by charges. ¯ kq qenclosed E ¯ E¯ dA¯ = E = 2 rˆ r · ✏0 I
... and charges feel a force in electric fields: F¯12 = qE¯
Magnetic fields, B ¯ , are created by moving charges. ¯ ¯ B µ0 Idl rˆ ¯ dB¯ = ⇥ B dr¯ = µ0Iencircled 4⇡ r2 · I ... and moving charges feel a force in magnetic fields: F¯ = qv¯ B¯ ⇥ Question...
q E¯ This tells us how charges interact with electric and magnetic fields B¯
But how do electric fields and magnetic fields interact with each other? Induced Currents In 1831...
Michael Faraday
Joseph Henry discovered: electric currents, I , arise (start) in circuits I in a changing magnetic field. B¯
This current is the induced current. Induced Currents 4 simple experiments: Experiment 1: moving a magnet near a coiled (loops) circuit
circuit has no battery (or power source)
move the magnet toward circuit and a current flows.
move the magnet away from circuit and an opposite current flows. Induced Currents 4 simple experiments: Experiment 1: moving a magnet near a coiled (loops) circuit
No movement = no current v =0 I =0 Induced Currents 4 simple experiments: Experiment 1: moving a magnet near a coiled (loops) circuit
Move the magnet and a current starts in the loop. v =0 I =0 6 6 (but no battery!) Induced Currents 4 simple experiments: Experiment 1: moving a magnet near a coiled (loops) circuit
Move the magnet faster and current increases. Induced Currents 4 simple experiments: Experiment 1: moving a magnet near a coiled (loops) circuit
Reverse direction of magnet current reverses direction Induced Currents 4 simple experiments: Experiment 2: moving a coiled circuit near a magnet
It doesn’t matter if the magnet moves or the circuit. Only relative motion is important. Induced Currents 4 simple experiments: Experiment 3: moving a current near a circuit Replace magnet with 2nd circuit
battery I
Steady current, I Circuit moves Induced current creates B¯ field. starts Induced Currents 4 simple experiments: Experiment 4: varying current in one circuit
When neither circuits moves; no induced current
Brief (short time) But, open switch current drops in left circuit current induced I 0 in right circuit ! B 0 ! Brief current induced Close switch in current increases in right circuit left circuit 0 I 0 B (opposite direction) ! ! Induced Currents B ¯ changes because ...
magnet moves towards circuit
circuit moves towards magnet
circuit creating magnetic field moves
magnetic field increases / decreases Induced Currents
Why B ¯ changes is not important.
A current appears in a circuit feeling a changing magnetic field.
This is electromagnetic induction. Induced Currents Quiz
Which will cause an induced current in a coil of wire?
(A) A magnet resting near the coil.
(B) The constant field of the Earth passing through the coil.
(C) A magnet being moved into or out of the coil.
(D) A wire carrying a constant current near the coil. The EMF Because the magnitude of current depends on the wire material, when describing induced current we talk about EMF.
EMF, : electromotive force E Potential difference ( V ) needed to cause a current to flow.
= V = E¯ dr¯ = IR (Ohm’s Law) E circuit · I lecture 7 around circuit material’s resistance; how easy it is to create current The EMF can be: e.g. a battery e.g. the induced EMF from induction
Changing B ¯ field creates an EMF. Magnetic flux
For electromagnetic induction, the surface is the inside of the circuit.
B ¯ is stronger closer to the magnet. Flux increases through circuit when magnet is close.
For a flat surface and uniform field: = B¯ A¯ = BAcos ✓ B ·
Unit: Tm2 or Wb (weber) Magnetic flux Quiz
A circular loop with 5.0 cm diameter makes an angle of 30 to uniform magnetic field, B¯ = 80mT . | | What is magnetic flux, B ?
4 (A) 1.6 10 Wb = B¯ A¯ = BAcos ✓ ⇥ B · 3 2 2 3 = (80 10 )⇡(2.5 10 ) cos(30 ) (B) 1.7 10 Wb ⇥ ⇥ ⇥
4 (C) 1.4 10 Wb ⇥
(D) 1.3Wb Magnetic flux Magnetic flux through loop next to current carrying wire.
B ¯ field encircles wire. Points into the page in the loop: B¯ dA¯==BdABA · µ I Field around a wire: B = 0 (lecture 8) 2⇡r varies with distance
a+W µ I µ Il a+w dr = BdA = 0 ldr = 0 B 2⇡r 2⇡ r Z Za Za µ Il a + w = 0 ln 2⇡ a ✓ ◆ Faraday’s Law
Using magnetic flux, B , and the EMF, , we can write: E
Faraday’s law
d B change in magnetic flux = E dt through circuit area induced EMF
The induced EMF opposes the flux change.
Describes electromagnetic induction in circuits. (more general form later) Faraday’s Law
d = B E dt
= B¯ A¯ = BAcos ✓ B · Faraday’s Law Example 1: changing B¯ Loop of radius r = 10 cm, resistance R = 2⌦ Uniform B ¯ increasing at 0.1T/s Find magnitude of induced current, I .
2 Ring is perpendicular to field: B = BA = B⇡r
d B d dB = (B⇡r2) = ⇡r2 = ⇡(0.1)20.1 dt dt dt
radius constant d = B = 3.14mV E dt 3.14mV I = |E| = =1.6mA R 2.0⌦ Faraday’s Law Example 2: changing area Sliding conducting bar changes circuit area.
Constant B ¯ : B = BA= Blx
d dx B = Bl = Blv dt dt d = B E dt
Blv I = |E| = R R Faraday’s Law Quiz A loop of area 240cm 2 and resistance R = 12 ⌦ carried an induced current I = 320mA . What rate is the magnetic field changing? dB dt
(A) 160T/s d /dt I = |E| = | B | R R (B) 0.67T/s d(BA)/dt dB/dt = | | = A| | (C) 0.016T/s R R
dB IR (0.32A)(12⌦) = = = 160T/s (D) 3.84T/s dt A 240 10 4m2 ⇥ Lenz’s Law Magnet moves Where did the energy come from? Induced current
work in moving the Loop heats magnet
But, movement at constant speed doesn’t take work ... unless there is a force opposing the movement.
The induced current must cause an opposing magnetic force.
Gives the direction of the current. Lenz’s Law d We knew this from Faraday’s Law: = B E dt The induced EMF opposes the flux change.
But it’s easier to find current direction from conservation of energy.
Ask: What direction will make it hard to move the magnet? Lenz’s Law When magnet is pushed into loop:
Want loop to make N pole: B¯ points left
Right-hand rule gives current direction.
When magnet is pushed out of loop:
Want loop to make S pole: B¯ points right
Right-hand rule gives current direction. Lenz’s Law
Current direction to oppose magnet motion.
Lenz’s Law:
The direction of an induced EMF or current is such that the magnetic field created by the induced current opposes the change in magnetic flux that created the current.
(current direction opposes motion) Lenz’s Law Quiz What will be the direction of the current in the loop when it enters the field, coming from the left?
(A) clockwise B¯ into page
(B) anti-clockwise Lenz’s Law Quiz Rectangular loop near a long wire.
a =1.0cm µ0Il a + w = ln B 2⇡ a w =3.5cm ✓ ◆ l =6.0cm Current in long wire increases at 25A / s . R = 50m⌦ 7 2 µ =4⇡ 10 N/A 0 ⇥ What is the induced current in the loop? (A) 18.0µA (C) 0.45µA
(B) 0.9µA (D) 9.0µA Lenz’s Law Quiz Rectangular loop near a long wire.
a =1.0cm µ0Il a + w = ln B 2⇡ a w =3.5cm ✓ ◆ l =6.0cm Current in long wire increases at 25A / s . R = 50m⌦
d /dt µ0l(dI/dt) a + w I = |E| = | B | = ln 2⇡R a R R ✓ ◆ 7 2 (4⇡ 10 N/A )(6.0cm)(25A/s) 4.5cm = ⇥ ln =9.0µA 2⇡(50m⌦) 1.0cm ✓ ◆ Lenz’s Law Quiz
Rectangular loop near a long a =1.0cm wire. w =3.5cm Current in long wire increases l =6.0cm at 25A / s . R = 50m⌦
What is the direction of the induced current?
(A) clockwise
(B) anti-clockwise Electric Generators Mechanical energy rotates loop
✓ changes B changes ! ( B = BA cos ✓ )
Creates alternating current d = B = IR E dt
Lenz’s law makes it hard to turn the loop.
Must burn fossil fuels (e.g. coal) or use nuclear fission to power the generator. Electric Generators Electric Generators An electrical generator has a 100-turn loop with diameter 50 cm. Rotated at frequency, f = 60rev / s to produce 60 Hz current. Find B ¯ for peak output voltage of 170 V (Produces actual peak of 120 V: normal in USA)
= B¯ A¯ = BAcos ✓ 1turn · = B⇡r2 cos ✓
Constant rotation: ! =2⇡f ✓ =2⇡ft ! 2 total = NB⇡r cos(2⇡ft) (N = 100 turns) Electric Generators An electrical generator has a 100-turn loop with diameter 50 cm. Rotated at frequency, f = 60rev / s to produce 60 Hz current. Find B ¯ for peak output voltage of 170 V (Produces actual peak of 120 V: normal in USA)
d = total E dt d = NB⇡r2 [cos(2⇡ft)] dt = NB⇡r2[ 2⇡f sin(2⇡ft)] Peak value: =2⇡2r2NBf = 170V B = 23mT Emax Electric Generators Quiz If power, P = I2R and you lower the electrical resistance (R) while turning the generator at constant speed, how will turning change?
Constant speed = constant change in ✓ (A) It will get harder = constant d B dt (B) It will get easier = constant EMF, E = IR E (C) No change if R drops, I must increase.
Therefore power increases Mutual Inductance As before (experiment 4)
A changing I in the left-hand coil
Creates a changing flux in right- hand coil
Induces a current in the right-hand coil
This is called mutual inductance.
The strength of the induced current depends on the flux that passes through the second coil. Self Inductance But the flux from a changing current also passes through its own circuit.
This induces an EMF that opposes the current change.
This makes it harder to increase the current.
This is called self inductance.
Self inductance is often a problem in circuits, but sometimes it is needed.
An inductor is designed to show self inductance. e.g. used to select specific frequencies by resisting change. Self Inductance Self inductance is defined as: L = B [Tm2/A] or H I (henry) d dI Differentiating: B = L dt dt
dI Giving Faraday’s law: = L (inductor EMF) EL dt
Since the inductor EMF opposes the current change, it is often called the back EMF. dI If is very large, the back EMF is very large. dt This can be very dangerous. Self Inductance A solenoid: cross-sectional area A length l n turns per unit length
What is self-inductance, L?
For a solenoid: B = µ0nI
Field perpendicular to coils: 1turn = BA
B = nlBA 2 =(nl)(µ0nI)A = µ0n AlI L = B = µ n2Al (inductance in a solenoid) I 0 Self Inductance A 5.0 A current is flowing in a 2.0 H inductor.
The current is reduced to zero in 1.0 ms.
Find the inductor EMF. dI 5.0A = L = (2.0H) = 10, 000V EL dt 1.0ms ✓ ◆ Potentially lethal!
Positive EMF: EMF increase in the same direction as the current. Tries to keep current flowing. Self Inductance Quiz A 2.0 A current is flowing in a 20 H inductor. A switch opens, stopping the current in 1.0 ms. What is the inducted EMF?
dI (A) 2.0kV = L EL dt
(B) 20kV = (20H)(2.0 103A/s) = 40kV ⇥
(C) 40kV
(D) 30.0kV Induced Electric Fields The induced EMF acts on static charges to produce a current.
Therefore, it is an electric field.
From earlier: = E¯ dr¯ work per unit charge gained E · I around circuit
d Faraday’s Law: E¯ dr¯ = B · dt I works for any closed loop Induced Electric Fields Faraday’s law tells us that electric fields have 2 sources:
Static charges
Changing magnetic fields
Field lines for induced electric fields have no start or end:
form closed loops encircling regions of changing magnetic field. Induced Electric Fields Solenoid: current increasing
field strength: B = bt constant
Find the induced electric field outside the solenoid at a distance r from the centre. (Note, very similar method to Ampere’s law)
d B 2 Faraday: E¯ dr¯ = B = BA = bt⇡R · dt I d B = ⇡R2b 2⇡rE dt R2b E = 2r Induced Electric Fields Quiz The induced electric field 12 cm from the axis of a 10 cm radius solenoid is 45 V/m.
What is the rate of change of the solenoid’s magnetic field? d E¯ dr¯ = B · dt (A) 0.75T/ms I 2⇡rE 2 (B) 0.6T/ms B = BA = ⇡R B d dB B = ⇡R2 (C) 0.06T/ms dt dt dB 2r E 2(12cm)(45V/m) = | | = (D) 1.1T/ms dt R2 (10cm)2 Diamagnetism A simple model for diamagnetism:
In a non-ferromagnetic material, atomic current loops point in different (random) directions.
With no magnetic field, the dipoles cancel.
In a changing magnetic field, the induced current causes one current direction to increase, the other to decrease.
Non-zero net magnetic moment opposing the field. Diamagnetism A superconductor is a perfect diamagnet. The induced currents completely cancels the applied magnetic field.
Inside the superconductor: B =0
The magnetic field is excluded from the superconductor.
This is called the Meissner effect.
The repulsive force between the superconductor and magnet causes magnetic levitation. Diamagnetism